Logarithmic exponential sums with multiplicative coefficients

Let $f$ be a $1$-bounded multiplicative function, and suppose that $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le 1\mathrm{/}(qx)$, where $(a,q)=1$ and $q\le x$. We prove that $$\mid \sum _{n\le x}\frac{f(n)}{n}e(n\alpha )\mid \le (\frac{4}{\sqrt{6}}+o(1))\frac{\log x}{\sqrt{q}}+O(\log q)\mathrm{.}$$ Here the constant $4\mathrm{/}\sqrt{6}$ is optimal. This settles two conjectures of Granville and Lamzouri recently. If $f$ is completely multiplicative, the constant improves to $1$. The main new ingredient is a two-prime Ramaré decomposition identity.